**Logarithmics Differentiation**

We next show how to use the logarithm as an aid to differentiation. The key idea is that if *F* is a function taking positive values then we can exploit the formula

**Example 1**

Calculate the derivative of the function

*F* ( *x* ) = (cos *x* ) ^{(sin x )} , 0 < *x* < .

**Solution 1**

We take the natural logarithm of both sides:

Now we calculate the derivative using the formula (*) preceding this example: The derivative of the left side of (†) is

Using the product rule, we see that the derivative of the far right side of (†) is

We conclude that

Thus

**You Try It** : Differentiate log _{9} |cos *x* |.

**You Try It** : Differentiate 3 ^{sin(3 x )} . Differentiate *x* ^{sin 3 x} .

**Example 2**

Calculate the derivative of *F* ( *x* ) = *x* ^{2} · (sin *x* ) · 5 ^{x} .

**Solution 2**

We have

Using formula (*), we conclude that

hence

**You Try It** : Calculate ( *d/dx* )[(ln *x* ) ^{ln x} ].

Find practice problems and solutions for these concepts at: Transcendental Functions Practice Test.

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